i.e., the usual homotopy intersection number, and the orientation of the joint path \(\xi \ast \xi _n\)  is
its positive time direction under homotopy with fixed end points, where the path \(\xi _n(t) = ({\begin{matrix}2-\frac{t}{\tau } & 0 \\ 0 & (2-\frac{t}{\tau })^{-1}\end{matrix}})\) . When \(\xi (2\pi )\in {\rm Sp}(2n)_{{\omega }}^0\) , the index \(i_{{\omega }}(\xi )\)  follows [1]}. The pair
\((i_{{\omega }}(\xi ), \nu _{{\omega }}(\xi )) \in {\bf Z}\times \lbrace 0,1,\ldots ,2n\rbrace \)  is called the index function of \(\xi \) 
at \({\omega }\) . When \(\nu _{{\omega }}(\xi )=0\)  or \(\nu _{{\omega }}(\xi )>0\) , the path \(\xi \)  is called \({\omega }\) -non-degenerate
or \({\omega }\) -degenerate respectively.
For more details readers may refer to [1]}.
